Question: Cory has $3$ apples, $2$ oranges and $2$ bananas. If Cory eats one piece of his fruit per day for a week and the pieces of fruit within each category are indistinguishable, in how many orders can Cory eat the fruit? One such order is $AAAOOBB.$
Solution: There are a total of $7$ fruits, so there are $7!$ ways to arrange them. However, since the fruits in each category are indistinguishable, we must divide out the repeats: $$\frac{7!}{3!2!2!}=7\times6\times5=\boxed{210}.$$